Problem 6 - Precalculus Functions and Graphs

Question

Fill in the blank. The ______________ of the function \(f(x)=\log _{a}(x)\) is \((0, \infty)\).

Step-by-Step Solution

Answer

The domain of the function \(f(x) = \log_{a}(x)\) is \((0, \infty)\).

Step 1: Understanding Logarithmic Functions

The function provided is a logarithmic function: \(f(x) = \log_{a}(x)\). Logarithmic functions have specific domains that need to be understood.

Step 2: Identify the Domain of Logarithmic Functions

For the function \(f(x) = \log_{a}(x)\), the variable \(x\) must always be greater than 0. Therefore, the domain of this function is the set of all positive real numbers.

Step 3: Fill in the Blank

Using the information from the previous steps, fill in the blank with 'domain'. The correct sentence is: The domain of the function \(f(x) = \log_{a}(x)\) is \((0, \infty)\).

Key Concepts

Domain of a Function

The domain of a function is a crucial concept in mathematics. It refers to the set of all possible input values (or 'x' values) for which the function is defined. For different functions, these input values can vary widely. When we talk specifically about logarithmic functions such as \(f(x) = \log_{a}(x)\), understanding the domain becomes even more important.

For a logarithmic function like \(f(x) = \log_{a}(x)\), the input value \(x\) must always be greater than zero. This is because logarithms are only defined for positive real numbers. Consequently, the domain of \(f(x) = \log_{a}(x)\) is the set of all positive real numbers, which we can write in interval notation as \(0, \infty\).

So, remember, when dealing with logarithmic functions: always check the domain to ensure all input values are valid!

Logarithmic Properties

Logarithmic properties are essential for understanding how logarithms work and how to manipulate them efficiently. Here are some fundamental properties:

• The logarithm of a product: \(\log_{a}(xy) = \log_{a}(x) + \log_{a}(y)\)
• The logarithm of a quotient: \(\log_{a}(\frac{x}{y}) = \log_{a}(x) - \log_{a}(y)\)
• The logarithm of a power: \(\log_{a}(x^{b}) = b \log_{a}(x)\)
• The change of base formula: \(\log_{a}(x) = \frac{\log_{b}(x)}{\log_{b}(a)}\)

Understanding these properties allows you to simplify complex logarithmic expressions and solve logarithmic equations more efficiently.

For example, if you wish to solve \(\log_{a}(x^{2})\), using the logarithm of a power property, you can rewrite it as \(2\log_{a}(x)\), making it easier to handle. Always refer back to these properties when working with logarithms to make your calculations smoother and more accurate.

Real Numbers

Real numbers include a wide range of numbers that are used in everyday math. These encompass all the numbers that can be found on the number line. Real numbers include:

• Natural numbers: 1, 2, 3, ...
• Whole numbers: 0, 1, 2, ...
• Integers: ..., -2, -1, 0, 1, 2, ...
• Rational numbers: fractions like 1/2, 3/4 or repeating decimals like 0.333...
• Irrational numbers: decimals that do not repeat or terminate, like π (pi) and √2

When we talk about the domain of logarithmic functions being \(0, \infty\), we refer specifically to positive real numbers. This means any positive number, whether whole, fraction, or irrational, is included.

Understanding real numbers is fundamental as they are used to describe almost any quantity in real life. They can represent distance, time, temperature, and more. So, having a good grasp of real numbers and their different types helps greatly in all areas of mathematics.